Clock drift, sampled exposure and robust mean-rate inference

Question and conditional result

The four-reading result separates necessary gain and clock intervals under a held clock rate. It does not test whether an unobserved clock is constant between readings. Here the complete structural law and held sensor are unchanged, while the clock rate belongs to a specified positive class with bounded derivative. The target is the first-window mean rate, not an instantaneous rate or the clock's complete time profile.

Two distinct results apply. Equal integrated clock exposures give exactly equal sampled full states, including the declared phase events. This obstructs profile and instantaneous-rate identification from these samples. Separately, a finite comparison bound transfers uncertain-clock readings to a constant-mean reference and permits reuse of the existing necessary inverse. The added intervals represent clock-model discrepancy; the physical sensor-error allowance is unchanged.

The theoretical results and implementation controls require no reserved response. The separately specified evaluation below tests their finite software application; earlier frozen protocols, sources and results remain unchanged. The ontology continues to separate collective observation from physical identification.

F1-F2: one structural law and a declared clock class

Retain the complete source and law: the eighteen-node two-port C9 support, degree matrix \(M\), \(A=M^{-1}L\), \(q=e_4-e_5\), held unit capacities and

\[ \frac{dx}{d\tau}=-Ax+\gamma f(\theta),\qquad \frac{d\theta}{d\tau}=\gamma Ax, \qquad \gamma=\frac1{1023\pi}. \tag{1} \]

The full original source has arbitrary common means and centered residuals \(u,v\) satisfying \(\|u\|_M\le X\), \(\|v\|_M\le Y\). The affine phase geometry, its original acute guard, the nominal priors \(b\in[11/8,3/2]\), \(c\in[2/3,1]\), and the actual original statistic \(B_{\rm initial}=b-(v_1-v_0)/8\) retain their existing owners. All thirty-six coordinates evolve; no source recentering or reset is added.

On the observation interval \([0,2H]\), declare

\[ \rho\in C^1([0,2H]),\qquad 0<\rho_-\le\rho(s)\le\rho_+,\qquad |\rho'(s)|\le\Lambda, \tag{2} \]

with \(H>0\), \(\Lambda\ge0\), and \(h_*:=\rho_+H\le1/2\). The independently supplied bound \(\Lambda\) has units of structural time per squared observation time. Define

\[ \tau(s)=\int_0^s\rho(v)\,dv,\qquad \bar\rho_1:=\frac{\tau(H)}H\in[\rho_-,\rho_+]. \tag{3} \]

Both rows of (1) multiply by \(\rho(s)\) in observation time. This follows the whole-law clock contract. The model class is a premise, not a profile estimated from the evaluated response. No clock samples or cached admission report substitute for it.

For \(0<a_1\le a_2\le1\), the phase jumps are \(a_1q\) at \(s=0\) and \((a_2-a_1)q\) at \(s=H\), with form continuous. There is no event at \(H/2\). At the same four times

\[ (s_0,s_1,s_2,s_3)=(0,H/2,H,2H),\qquad r_i=Gq^Tx(s_i)+O+\eta_i,\quad |\eta_i|\le\delta, \tag{4} \]

use one held positive gain \(G\in[G_-,G_+]\) and one held offset \(O\). The readings are instantaneous passive samples with primitive intervals \([\ell_i,u_i]\). Averaging over finite sensor apertures would be a different observation law. Input calibration, support, capacity and the structural coefficients are all held; no other nuisance law is relaxed.

F3-F4: exact dependence on integrated exposures

Let \(\Phi_t\) denote the autonomous full flow of (1), and let \(\mathcal J_a\) add \(aq\) to phase without changing form. Define the three structural exposures

\[ e_1=\tau(H/2),\qquad e_2=\tau(H)-\tau(H/2),\qquad e_3=\tau(2H)-\tau(H). \tag{5} \]

Since \(\rho>0\), the chain rule and uniqueness give

\[ \begin{aligned} z_{1/2}&=\Phi_{e_1}\mathcal J_{a_1}z^-,\\ z_{H^-}&=\Phi_{e_2}z_{1/2},\qquad z_{H^+}=\mathcal J_{a_2-a_1}z_{H^-},\\ z_{2H}&=\Phi_{e_3}z_{H^+},\qquad z=(x,\theta). \end{aligned}\tag{6} \]

Thus the complete sampled history factors through the three exposures. Two clocks with the same \((e_1,e_2,e_3)\), source and observed-time event schedule give exactly the same sampled full states. With the same held sensor and allowed error realization, they give the same readings. This is a sufficient equality condition; distinct exposures may also be indistinguishable through a partial observation or a special trajectory. It does not prove that the three exposures themselves are identified.

An explicit smooth obstruction works for every \(\Lambda>0\) and nondegenerate rate prior. Choose an interior \(\rho_0\), and

\[ \rho_\epsilon(s)=\rho_0+\epsilon\cos(4\pi s/H),\qquad 0<\epsilon<\min\{\rho_0-\rho_-,\rho_+-\rho_0, \Lambda H/(4\pi)\}. \tag{7} \]

This clock satisfies (2). Each interval in (5) contains an integer number of its cosine periods, so its exposures equal those of the constant clock \(\rho_0\). Nevertheless \(\rho_\epsilon(s_i)=\rho_0+\epsilon\) at every reading time, whereas the constant clock's rate is \(\rho_0\). Even noiseless full-state samples at those times cannot distinguish their instantaneous rates or verify clock constancy. Their first-window mean is the same, so this obstruction does not preclude bounds on \(\bar\rho_1\).

The equivalence relies on an autonomous structural law, events at the fixed observation times and instantaneous samples. A different event selector, structural-time forcing or observation aperture requires its own proof. Between the sampled times the full observed-time histories generally differ.

A finite comparison with an exactly matching event state

Use a reference history with the same full source, law, inputs, gain, offset and errors, but constant rate \(\bar\rho_1\). It agrees with the actual history at \(0\) and \(H\), because both have elapsed structural time \(\tau(H)=H\bar\rho_1\) before the second event. Their complete post-event states therefore agree as well. No stability approximation or inferred intermediate state is needed at that jump.

Set \(\Delta\rho=\rho_+-\rho_-\). The two possibly nonzero exposure mismatches satisfy

\[ \begin{aligned} |\tau(H/2)-\tau(H)/2|&\le d_{1/2}:=\min\{\Lambda H^2/8,\Delta\rho H/4\},\\ |\tau(2H)-2\tau(H)|&\le d_2:=\min\{\Lambda H^2,\Delta\rho H\}. \end{aligned}\tag{8} \]

For the first bound, write the difference as \(\frac12\int_0^{H/2}[\rho(v)-\rho(v+H/2)]\,dv\). For the second, use \(\int_0^H[\rho(v+H)-\rho(v)]\,dv\). The derivative bound and global rate range give the two respective bounds inside each minimum. Linear clocks \(\rho(s)=\rho_0+\kappa s\) have exact signed differences \(-\kappa H^2/8\) and \(\kappa H^2\), so the derivative-based coefficients cannot be uniformly reduced when such clocks are admitted by the rate range.

The following global bound avoids duplicating or reevaluating the inverse's specialized source remainder. Let \(g=1/3069>\gamma\). On this undirected support, \(A\) is positive semidefinite and self-adjoint in the \(M\) inner product, so \(e^{-tA}\) is a contraction. Pairwise sine currents have zero degree-weighted mean and \(\|f(\theta)\|_M\le\sqrt{40}<7\) for every phase state. Variation of constants, unchanged form at the phase events, and \(\tau(2H)\le2h_*\) consequently give

\[ \|P_Mx\|_M\le Q_0:=X+14gh_*,\qquad \left\|\frac{dx}{d\tau}\right\|_M \le U:=2Q_0+7g. \tag{9} \]

The bound includes both continuous rows: phase evolves throughout and its sine forcing is bounded globally, not held at its initial value. Arbitrary common form and phase means do not enter it. The same bound holds along the reference and along the comparison arcs between their sampled points, because each such arc lies on the same pre-event or post-event full flow with total structural elapsed time at most \(2h_*\).

Since \(\|q\|_{M^{-1}}=1\), the actual/reference recorded differences, coupled with the same four sensor errors, obey

\[ |r_i-r_i^{\rm ref}|\le b_i,\qquad (b_0,b_1,b_2,b_3)=(0,G_+U d_{1/2},0,G_+U d_2). \tag{10} \]

Consequently every actual compatible history induces reference readings in

\[ \widetilde I_i=[\ell_i-b_i,u_i+b_i].\tag{11} \]

These are necessary comparison bands. They are not new measured readings, independent clock errors or independent sensor noise. Their midpoint is unchanged, and the exact common sensor offset still cancels. In particular, the first whole-window increment is unchanged, the second changes by at most \(b_3\), and the finite curvature changes by at most \(2b_1\).

The instantaneous chain-rule expression makes the same scope visible:

\[ \frac{d^2}{ds^2}(Gq^Tx) =G\rho(s)^2q^Tx_{\tau\tau}+G\rho'(s)q^Tx_\tau. \tag{12} \]

The second term must not be silently assigned to structural curvature. Equations (8)-(11) bound its finite-reading consequence without supplying observed derivatives. After dividing a second difference by \(H^2\), the derivative-based discrepancy need not vanish as \(H\) decreases. A shorter observation window alone does not remove clock uncertainty.

Necessary mean-rate inference through one shared reference envelope

Re-admit the eleven primitives, including the original four reading pairs, positive global rate/gain priors and nonnegative \(\Lambda\). Compute (8)-(11) with normalized exact scalars before interval materialization. Invoke the curvature inverse once with the comparison bands, original \(\delta\), observed duration \(H\) and otherwise unchanged source/input priors. Its unknown held clock now denotes \(\bar\rho_1\), and its effective gain is

\[ J_1=G\bar\rho_1.\tag{13} \]

Every actual history in (1)-(4) produces a compatible constant-mean reference, so every necessary reference marginal contains the actual original geometry, held gain and first-window mean. A strict reference exclusion rules out all actual histories satisfying these premises. The converse does not follow: marginal interval intersection discards correlations among source, clock exposures and reading errors and need not describe a jointly realizable history.

The parent report explicitly names first_window_mean_clock_rate_outer_bounds. Its nested reference_envelope is an auxiliary necessary calculation, not a claim that the actual clock was constant. The original readings and sensor error remain separate from the exact comparison bands and transfer allowances. The source and original-angle meaning remain unchanged. No instantaneous rate, clock profile or verification of the supplied derivative bound is returned.

bounded_candidate means admitted necessary marginals; incompatible means strict necessary exclusion; unavailable retains the child method's source, rank or arithmetic limitation. A wide available interval is not a proof of nonidentifiability. Global transfer bounds are candidates until the report's original-source admission supplies their inference scope. If \(\Lambda=0\), or the global rate prior is a singleton, both exposure allowances vanish and the comparison child is exactly the existing constant-clock calculation on the original readings.

An informative positive-drift regime without a response evaluation

Retain the previous finite-error budget and add a strictly positive derivative allowance:

\[ \begin{gathered} H=2^{-24},\quad X=Y=2^{-48},\quad (a_1,a_2)=(1/4,3/4),\\ G\in[1,2],\quad\rho(s)\in[1/2,2],\quad \delta=2^{-90},\quad t_i\le\delta,\quad\Lambda=2^{-22}. \end{gathered}\tag{14} \]

Here \(t_i\) are the original primitive reading half-widths. No clock realization, source realization or reading is selected. Let \(R_*\) be the previous radius bound for the two leading inverse coordinates at \(h_*=2H\). Only the final reading's extra allowance enters that three-reading inverse. The inherited column norm is at most \(36000/h_*\), so a sufficient radius is

\[ R_{\rm drift}=R_*+36000b_3/h_*<1/12500. \tag{15} \]

The original first-quadrant and positive-radius arguments therefore give

\[ w_{J_1}\le4R_{\rm drift}<1/3000,\qquad w_b\le\frac{2R_{\rm drift}}{1/4-2R_{\rm drift}}<1/1500. \tag{16} \]

The reference's complete finite-curvature remainder \(D_C\) is unchanged and remains below \(10^{-9}\). Its normalized numerator diameter now satisfies

\[ d_A\le\frac{64\delta+16b_1}{H^2}<2^{-28}. \tag{17} \]

Reuse the positive natural coefficient bound \(K_->k_0=1/18000\) and \(w_K\le11g w_b/8\), with the same proof and priors. Put \(\beta=w_K/k_0\) and \(t=2w_{J_1}+\beta+2w_{J_1}\beta\). Compatibility still guarantees positive quotient numerators. The same endpoint-width argument yields

\[ \begin{aligned} w_{\bar\rho_1}&\le 2d_A/k_0+(2+D_C/k_0)t+2D_C/k_0<1/80,\\ w_G&\le2w_{J_1}+4w_{\bar\rho_1}<1/16. \end{aligned}\tag{18} \]

The strict bounds follow by rational substitution; the last inequalities also hold with the conservative allowances in (16)-(17) and \(D_C<10^{-9}\). Original actual-angle width remains at most \(w_b+Y/4\). These are conditional exact-arithmetic budgets, not new observations, a physical sensor specification or an evaluated drift trial. Executable outward arithmetic and inverse availability remain separate checks. The positive drift class still contains the exact ambiguity (7).

Implementation, evidence and research boundary

The clock-drift adapter uses shared primitive admission, one fresh curvature inverse and the global comparison bound (9). It introduces no second inverse solver, profile fit, trajectory generator or cached-report admission. The contract owns input/output semantics; the controls cover exact clock limits, exposure bounds, original-error association and fresh complete-flow comparisons separately from frozen evidence.

The result does not infer a supplied clock law, verify its derivative allowance from four samples, or identify physical time units. Common all-row-law-rate/clock equivalence also remains if that additional unknown is admitted. Source formation, support selection, work supply and future maintenance retain their separate obligations. The sole execution plan owns any subsequent admission; no previous frozen response is replayed or reinterpreted under this larger clock class.

Prospective software evaluation with nonconstant clocks

This protocol evaluates the finite-transfer method on new complete-flow responses. Its preparation, clocks, sensor, numerical budget and stopping criteria are fixed before the first response. The existing held-clock records are not inputs to this evaluation. The exact-exposure theorem permits reuse of the unchanged structural-time producer; no new variable- time solver or approximate clock quadrature is introduced.

F1-F2: shared preparation and three declared histories

Use the public budget (14), including the positive allowance \(\Lambda=2^{-22}\), and the same public geometry priors. Set the hidden original geometry to \(b=23/16\), \(c=4/5\), the held sensor gain to \(G=7/5\), and the target first-window mean to \(\bar\rho_1=9/8\). These are supplied software preparations, not parameters selected from an evaluated response. Both variable-clock cases share their entire original source and sensor, with offset \(O=5/13\) and the same four errors

\[ (\eta_0,\eta_{1/2},\eta_1,\eta_2)=2^{-91}(-1,1,-1,1). \tag{19} \]

For the eighteen nodes, use residual recipe \(r=4\):

\[ \widetilde u_i=\frac{(i+2)(r+2)}{2^{62}},\qquad \widetilde v_i=\frac{(5i+3r)\bmod23+1}{2^{62}},\qquad u=P_M\widetilde u,\quad v=P_M\widetilde v. \tag{20} \]

The nominal phase geometry and its full weighted centering retain the source owner. Add common form \(6/11\), common phase \(-6/13\) and the corresponding centered residual to each coordinate. Retain the exact arrays and outward mathematical-pi source box. Every residual entry is nonzero; weighted centering preserves the squared-norm bound, and \(40\cdot114^2/2^{124}<2^{-96}\) admits both source norms. The original actual long-arc mean is \(b-5/2^{65}\), since the first two phase residuals differ by \(5/2^{62}\). Source-box materialization must separately retain the declared norm bounds and this statistic.

The two variable clocks and one shared constant-mean reference are

\[ \rho_\pm(s)=\bar\rho_1\pm\frac{\Lambda}{2}(s-H/2),\qquad \rho_{\rm ref}(s)=\bar\rho_1. \tag{21} \]

All satisfy the global prior and derivative allowance. Their first-window means agree exactly. For \(\kappa=\pm\Lambda/2\), their three structural segment durations are

\[ \left(\bar\rho_1H/2-\kappa H^2/8,\quad \bar\rho_1H/2+\kappa H^2/8,\quad \bar\rho_1H+\kappa H^2\right). \tag{22} \]

These are positive exact rational exposures. The reference uses the same expression with \(\kappa=0\). Thus both nonconstant cases have strictly nonzero half-time and final exposure differences from the reference. Both rows of the complete autonomous law undergo the same time change; the reduction is exact, rather than freezing phase or using the mean rate separately in every actual segment.

An additional ambiguity control is the declared profile

\[ \rho_{\rm companion}(s)=\rho_+(s) +\epsilon\cos(4\pi s/H),\qquad \epsilon=\Lambda H/32. \tag{23} \]

Its derivative has magnitude at most \(\Lambda/2+\pi\Lambda/8<\Lambda\), and its distance from \(\bar\rho_1\) is at most \(25\Lambda H/32\), well inside the fixed global prior. Its three exposures equal those of \(\rho_+\), while every sampled instantaneous rate differs by \(\epsilon>0\). By (6), the same complete trajectory record encloses its sampled states. Retain this explicit association and the analytic exposure certificate; do not rerun an identical producer or count the association as a second independent numerical response. No profile is inferred from a mean-rate output.

F3: exact exposure execution and separate public inference

Generate exactly three complete structural histories: the two clocks in (21) and their shared reference. Each uses the existing full-state producer with three sequential exposures (22), phase increments \((1/4,0,1/2)\), and all thirty-six endpoint intervals carried unchanged. The observation times remain \((0,H/2,H,2H)\). There is no event at the half-time sample and no state reset at either sample or the second phase jump.

Use one direct source-box order-four Taylor step per segment, its whole-tube fifth-order remainder, the fixed sixteen-attempt Picard budget, and shared outward dyadic-128 arithmetic. Retain all nine full certificates, their source associations, strict inclusion margins, achieved horizons and both global clock coordinates. Do not subdivide, increase precision, replace the source box by its midpoint or retry a failed budget. Each sensor reading must have numerical half-width at most \(\delta=2^{-90}\), separately from its physical-error allowance. Optional acute flags remain observations, not stopping requirements.

For each variable-clock case, pass only the eleven documented primitives to the isolated inverse worker. In particular, probe_duration is observed \(H\), and the clock inputs are the global prior and derivative bound. The source arrays, exact geometry, true gain, clock profiles, exposures, mean rate, reference trajectory and realized errors remain on the producer/evidence side. Retain allowlisted packets and hashes. The reference is used only for the independent transfer assessment, with no inverse evaluation. This is software information exclusion, not a secrecy or execution-authentication guarantee.

Response-free signed separation budget

The transfer controls have a prospective finite-flow sign, not merely different clock formulas. Set \(T=4H\), \(Q_0=X+7gT\) and \(V=Y+2gTQ_0\). Equation (9) and the phase row give, throughout either pulse window and all comparison arcs,

\[ \|P_Mx\|_M\le Q_0,\qquad \|\theta-\theta_{\rm nominal}-m_\theta\mathbf1-a_jq\|_M \le V.\tag{24} \]

The centered nominal phase uses the same source convention as (20). At its pulse-shifted value, \(q^Tf=-F_j\), where \(F_j=2\sin(3a_j/2)\cos(b-a_j/2)>1/6\). For \(a_1=1/4\), use \(\sin(3/8)>1/3\) and the alternating cosine lower polynomial at \(21/16\), which exceeds \(1/4\). For \(a_2=3/4\), use \(\sin(9/8)>1/2\) and \(\cos(17/16)\ge223/512>1/6\). The sine map has weighted Lipschitz bound two, \(\|A\|_M\le2\), and \(1/3216<\gamma<g\). Consequently

\[ q^Tx_\tau<-m,\qquad m:=\frac1{6\cdot3216}-2Q_0-2gV>\frac1{20000}. \tag{25} \]

All bounds follow from the declared primitives and elementary inequalities; no reserved response or measured derivative is used.

The common sensor errors cancel between histories. Write \(D_{\pm,i}\) for the true variable/reference recorded difference. The half-time and final signs are

\[ D_{+,1}>0>D_{-,1},\qquad D_{+,3}<0<D_{-,3}, \tag{26} \]

with half-time magnitudes exceeding \(G\Lambda H^2/(16\cdot20000)=(14336/3125)\delta>4\delta\), and final magnitudes exceeding eight times that amount. Exact equality at \(0,H\) gives \(C_- - C_+>(57344/3125)\delta>16\delta\). An individual recorded interval of radius at most \(\delta\) has width at most \(2\delta\). Thus its variable/reference difference can extend at most \(4\delta\) beyond the true difference, and the two curvature intervals' difference at most \(16\delta\). The frozen numerical half-width criterion therefore suffices to retain all signs and the strict curvature ordering in the recorded intervals.

For containment, the actual gain and half-budget slopes yield \(|D_{\pm,i}|\le(7/20)b_i\), for \(i=1,3\). The derivative cap is the active term in (8), and \(b_1/\delta=2^{18}U>500\), with \(b_3=8b_1\). Hence the additional \(4\delta\) enclosure allowance is less than \((13/20)b_i\). The entire retained difference intervals must fit strictly inside the transfer bands whenever their individual reading-width criteria pass. Inverse availability and the complete validated horizons still require the declared executable checks.

F4: fixed coverage, transfer and ambiguity criteria

The first evaluation passes only if all source/clock/sensor admissions, complete horizons, endpoint associations and numerical budgets pass, and:

• Both public inverses return bounded_candidate, with source, drift transfer, finite curvature, rank and positive-division evidence admitted. Their necessary marginals cover the original nominal and actual angle, the retained initial actual-angle box, \(J_1=63/40\), \(G=7/5\) and \(\bar\rho_1=9/8\).
• Strict widths are below 1/1024 for the actual original angle, 1/2048 for effective gain, 1/16 for gain and 1/80 for mean rate. Nonempty marginals do not imply joint realizability of every retained parameter tuple.
• Each variable-clock half-time and final recorded interval has the strict signed separation (26) from its reference interval. The entire difference interval lies in the corresponding \([-b_i,b_i]\) from (10). At \(H\), every pair of actual/reference full-state coordinate boxes intersects, before and after the common phase event. This is numerical consistency with the exact theorem, not an overlap proof of equality.
• The finite-curvature intervals have the strict ordering \(C_+<C_-\). Each uses the same four recorded readings and their original sensor errors, without an independently supplied derivative observation.
• False global clock prior [25/32,13/16] and false gain prior [31/16,2] leave the corresponding coarse child available but produce a strict curvature exclusion. The false angle prior [11/8,353/256] is excluded; equal cumulative amplitudes (1/4,1/4) give the inherited rank abstention. These controls use the same actual readings with altered declarations.
• Both full-window increments exclude the complete phase-blind alternative \(x_\tau=-Ax,\ \theta_\tau=\gamma Ax\), whose necessary recorded band remains \([-B_{\rm heat},B_{\rm heat}]\), with \(B_{\rm heat}=2G_+\rho_+HX+2\delta\). The varying clock still gives at most \(\rho_+H\) structural time per window.
• The companion clock's global and derivative bounds, strictly different sampled rates, equal exposures and association with the positive-slope complete record pass independently reconstructed exact checks.

Freeze this prospective text, the machine-readable protocol, committed runtime source, evaluator and public worker before any response. The exclusive attempt ledger precedes the first flow. Retain the first outcome, including partial responses, failed criteria, unavailability or exception; no budget adjustment or second attempt is admitted. Corrections require separate evidence and cannot replace that outcome.

Success would establish finite software robustness of the mean-rate/gain inference to the declared clock variation, and exercise a nonzero transfer allowance. It would not identify the instantaneous profile, establish physical clock units or feasible laboratory noise, select a unique law, or derive source formation and maintenance. The execution plan alone owns the next research boundary.

Retained first nonconstant-clock evaluation

The first reserved evaluation returned certified_reserved_clock_drift_inference: all 112 fixed conditions passed. The count comprises sixteen reference-history conditions, forty-three for each variable-clock history, one shared-source condition, one paired-curvature condition and eight companion-clock conditions. The protocol, source archive, attempt, response and manifest preserve that first outcome. The prospective statements above retain their pre-evaluation wording; no retry, altered budget or older producer replay was used.

All nine full thirty-six-coordinate segments completed their prescribed structural exposures with strict Picard inclusion. Every endpoint was carried unchanged into the next declared phase event or passive continuation. Both public inverse packets contained only the eleven documented primitives. Neither the reference history nor the companion profile entered those packets or received a separate inverse evaluation.

Both necessary marginal reports cover the original nominal angle, \(B_{\rm initial}=23/16-5/2^{65}\), its retained initial source box, \(J_1=63/40\), \(G=7/5\) and \(\bar\rho_1=9/8\). The following widths are decimal upper bounds, rounded upwards to twelve places from the stored exact rational widths. The stopping tests use the rational values and their frozen strict thresholds.

Clock history	Original-angle width	Effective-gain width	Gain width	First-window mean-rate width
Positive slope	0.000057836407	0.000212441681	0.000803235837	0.000493713258
Negative slope	0.000057836407	0.000212441681	0.000803235731	0.000493713316
Frozen strict threshold	1/1024	1/2048	1/16	1/80

The independently generated reference comparisons exercise nonzero transfer, rather than only exposure-equivalent clocks. In units of the supplied sensor allowance \(\delta\), the retained differences have the following outward-rounded enclosures:

Clock history	Half-time recorded difference / delta	Final recorded difference / delta
Positive slope	[5.341891,5.341892]	[-200.581040,-200.581039]
Negative slope	[-5.341892,-5.341891]	[200.581039,200.581040]

Every entire difference interval lies inside its independently rebuilt transfer band. Both histories have the predicted strict sign at each sample, and their recorded curvature intervals satisfy \(C_+<C_-\). At \(H\), all actual/reference coordinate enclosures intersect before and after the second event. The exact equality follows from the exposure theorem; the overlap is its numerical consistency check.

The false clock/gain priors leave their coarse children available but are excluded by the finite-curvature refinement. The false angle is also excluded; equal cumulative inputs produce rank-based unavailability; both full-window increments exclude the specified complete phase-blind law. These controls retain their prospective meanings and do not establish joint realizability of alternative marginal tuples.

The companion record reconstructs its positive global rate, derivative bound and integer cosine turns. Its sampled rates differ from those of the positive-slope history by \(\epsilon=2^{-51}\), yet its sample/event exposures agree exactly. It therefore uses the same full-state enclosure by theorem, with an explicit source/sensor/trajectory association. This is an exact ambiguity control, not a second independently simulated or measured response.

The result establishes finite software robustness of the mean-rate/gain inference within the frozen preparation, clock class and held observation law. The extremely small stipulated sensor error and clock variation are numerical protocol premises; no experimental sensor performance or clock calibration follows. Instantaneous-profile ambiguity, joint source/noise feasibility, physical units, formation and maintenance remain separate questions. The global smooth-flow domain flags establish no acute-sector or identity-retention certificate.

The source base is 1a717ca8f4f264f277bfdf661d9aeee4dd103af9, with no runtime overlays. Its archive SHA-256 is 1a08af6a46c091cb49b6a1da8d12186768b824470facf14db5131193500a10b0. The archived prospective owner contains 25,898 bytes, with SHA-256 f1cebe3ab44153e51d700dbc7aba6bbc270cf87b375d08a70b2df1d916e98ce7. Those bytes remain the unchanged prefix of this owner, modulo checkout newline conversion. Hashes associate retained records; they do not independently authenticate chronology, execution or physical acquisition. The read-only evidence audit rebuilds the exposure, transfer, certificate and inverse arithmetic without rerunning any producer, inverse or archived worker. The execution plan owns the closed gate and its separate inactive resumption boundary.